arxiv:1510.07833v2 [math.ca] 28 oct 2015arxiv:1510.07833v2 [math.ca] 28 oct 2015 a new definition of...

arX

iv:1

510.

0783

3v3

[m

ath.

CA

] 1

2 N

ov 2

019

A NEW DEFINITION OF ROUGH PATHS ON MANIFOLDS

YOUNESS BOUTAIB AND TERRY LYONS

Abstract. Smooth manifolds are not the suitable context for trying to generalize the con-cept of rough paths on a manifold. Indeed, when one is working with smooth maps insteadof Lipschitz maps and trying to solve a rough differential equation, one loses the quanti-tative estimates controlling the convergence of the Picard sequence. Moreover, even witha definition of rough paths in smooth manifolds, ordinary and rough differential equationscan only be solved locally in such case. In this paper, we first recall the foundations ofthe Lipschitz geometry, introduced in [8], along with the main findings that encompass theclassical theory of rough paths in Banach spaces. Then we give what we believe to be aminimal framework for defining rough paths on a manifold that is both less rigid than theclassical one and emphasized on the local behaviour of rough paths. We end by explaininghow this same idea can be used to define any notion of coloured paths on a manifold.

1. Introduction

The theory of rough paths (Lyons, [25]) and its variations (e.g. Gubinelli, [16]) generaliseYoung’s integration theory in a way that it separates the probabilistic and deterministicparts of the strongly probabilistic Ito calculus, using only the variation of paths as a way ofmeasuring their smoothness. A crowning achievement of the theory is understanding that,as far as ordinary differential equations are concerned, a path should not be defined by itsgraph but rather be identified as a choice of its signature (that is, the sequence of its iter-ated integrals). For example, the signature of a Brownian motion could be calculated usingeither Ito’s or Stratonovich’s calculus and it is this choice that leads one to solve a stochasticdifferential equation driven by a Brownian motion either in the sense of Ito’s calculus orStratonovich’s. More generally, the theory of rough paths provides us with a deterministiccalculus constructed in a way that it does not depend intrinsically on how a signature isdefined but rather on common algebraic (that can be summerized by a Hopf algebra struc-ture) and analytical properties that all signatures are expected to satisfy. These works have,in particular, enriched the toolbox of stochastic analysis with deterministic -path by path-results and widened its scope to rougher paths than the Brownian motion. The underlyingphilosophy of the theory also opened the door for solving a certain class of Stochastic PartialDifferential Equations that require making sense of classically ill-defined products of distri-butions. This was carried incrementally through the development of alternative rough paththeories: branched rough paths [17], para-controlled calculus [18] and regularity structures[20] (see [14] for a succinct exposition that goes from the theory of rough paths to that ofregularity structures.) In addition to the natural applications that come with stochastic

The first author carried most of this research while at the University of Oxford. The same author isthankful for the support of the DFG through the research unit FOR2402 in Berlin and Potsdam whereimportant improvements on this work have been made.

1

http://arxiv.org/abs/1510.07833v3

A NEW DEFINITION OF ROUGH PATHS ON MANIFOLDS 2

analyis, the theory of rough paths highlighted the role of signatures as highly efficient trans-forms of paths [4, 21], which led to their exploitation in recent works in machine learning (theliterature on the subject being abundant, we cite [2, 11, 19] as varied use-case examples).

While the need for such calculus on manifolds arises naturally from both the points ofview of pure and applied mathematics (see for example the introductions to [8, 23] for themotivation), the literature on rough paths on manifolds is still very limited compared to itscounterpart in the Euclidean setting or even to that of stochastic analysis on manifolds. Themain attempts to generalise these notions to manifolds are due to Cass, Litterer and Lyons in[8] in the framework of what is called a Lipschitz manifold and Driver and Semko in [13] (forpaths controlled by rough paths on Riemannian manifolds). We also refer to Cass, Driver andLitterer in [7] (for weakly geometric rough paths on submanifolds embedded in the Euclideanspace) and Bailleul [3]. As in [8], we aim to avoid to put too much structure on the manifoldwe work on or to exactly mimic the construction of rough paths on the Euclidean space inthe non-linear framework of a manifold. In order to show how one can simply translate thenotion of rough paths (and in general that of coloured paths) to manifolds, we use belowthe language of category theory to introduce local Lipschitz manifolds and explain how thiscan be done without any further particular considerations of the class of manifolds one isworking on (for example, one does not need the manifold to be Riemannian to be able tomeasure the smoothness of paths).

2. Review of key elements in the theory of the rough paths

We start by defining rough paths and giving a reminder of all the notions that will benecessary to us in the rest of this work, one of which will be the extension of the notion ofLipschitz (Holder) maps. Most of the results recalled in this section can be easily found,for example, in [6], [25] and [26]. We mostly give some examples and prove some of thestatements to familiarize the reader with any possible new notions. In particular, we willattach a certain importance to associating a starting point or a trace to geometric roughpaths.

2.1. Signatures of paths.

2.1.1. The concept of the p-variation. We introduce the notion of p-variation, crucial inboth Stieltjes’s and Young’s integration theories and the rough path theory. It is a way ofmeasuring the amplitude of the oscillations of a path, independently of when said oscillationsoccur.

Definition 2.1. Let p ≥ 1, (E, ‖.‖) be a normed vector space and T ≥ 0. Let x : [0, T ] → Ebe a continuous path. For a finite subdivision D = (ti)0≤i≤n of [0, T ], we denote by ‖x‖p,Dthe quantity:

‖x‖p,D :=

(n−1∑

i=0

‖xti+1− xti‖

p

) 1p

=

(∑

D

‖xti+1− xti‖

p

) 1p

x is said to have a finite p-variation over [0, T ] if ‖x‖p,D|D ∈ D[0,T ] has a finite supremum.In this case, this supremum is called the p-variation of x over [0, T ] and is denoted by‖x‖p,[0,T ]. When p = 1, we say that the path x has bounded variation.

Remark 2.2. The previous definition can easily be adapted for a metric space (E, d) but wewill not use this generalisation in the rest of these notes.


Examples 2.3. Let T ≥ 0:

(1) Let E be a normed vector space. A continuously differentiable E-valued path x over[0, T ] is of bounded variation over [0, T ] and ‖x‖1,[0,T ] ≤ T‖x‖∞,[0,T ].

(2) Let E be a normed vector space and 0 < α ≤ 1. An α-Holder E-valued path over[0, T ] has finite 1/α-variation over [0, T ].

(3) Let (Ω,A,P) be a probability space and B be a Brownian motion over [0, T ]:• B has finite p-variation over [0, T ] almost surely, for any p > 2.• Seen as an L2(Ω)-valued path, B has finite 2-variation.

Proposition 2.4. [6, 12] Let p ≥ 1 and (E, ‖.‖) be a Banach space.

• The set Vp([0, T ], E) of all continuous paths from [0, T ] to E that have a finite p-variation over [0, T ] is a vector space when endowed with the natural operations ofaddition and multiplication by a scalar.

• The map:

‖.‖Vp([0,T ],E) : Vp([0, T ], E) → R+

x 7→ ‖x‖p,[0,T ] + supt∈[0,T ]

‖xt‖

defines a norm on Vp([0, T ], E) called the p-variation norm.• (Vp([0, T ], E), ‖.‖Vp([0,T ],E)) is a Banach space.• ∀q ≥ p ≥ 1 : Vp([0, T ], E) ⊆ Vq([0, T ], E).

The manipulation of p-variations is often made easier by the introduction of controls. Fora compact interval J , we will denote by ∆J the simplex of all pairs (s, t) ∈ J2 such thats ≤ t.

Definition 2.5. A function ω : ∆[0,T ] → R+ is said to be a control if it has the followingproperties:

• ω is continuous.• ω is super-additive i.e. ω(s, u) + ω(u, t) ≤ ω(s, t) , ∀ 0 ≤ s ≤ u ≤ t ≤ T .

Lemma 2.6 shows that to every path of finite p-variation, we can associate a “natural”control.

Lemma 2.6. [6, 12, 26] Let p ≥ 1, E be a normed vector space and T ≥ 0. Let x : [0, T ] → Ebe a continuous path of finite p-variation over [0, T ]. Then the function ω defined on ∆[0,T ]

by ω(s, t) = ‖x‖pp,[s,t] is a control.

Conversely, if we can find a control that bounds from above the pth power of the incrementsof a continuous path then this path is necessarily of finite p-variation. This will be the contentof theorem 2.7. Consequently, it also gives a criterion for the finiteness of the p-variation ofa path without having to go back to the definition.

Theorem 2.7. [6, 12, 26] Let p ≥ 1, E be a normed vector space and T ≥ 0. Let x :[0, T ] → E be a continuous path. There exists a control ω defined over ∆[0,T ] such that forevery (s, t) ∈ ∆[0,T ] we have: ‖xt−xs‖

p ≤ ω(s, t) if and only if x has a finite p-variation. Inthis case, and for such a control ω, we have:

∀(s, t) ∈ ∆[0,T ] : ‖x‖pp,[s,t] ≤ ω(s, t)

We say in this case that the p-variation of x is controlled by ω.


2.1.2. The signature of a path. We start by recalling the definition of the tensor algebra ofa vector space (see [28] for an exhaustive exposition).

Definition 2.8. Let E be a vector space. For every n ∈ N∗, let E⊗n be the space of homoge-neous tensors of E of degree n. We use the convention: E⊗0 = R. The set of formal seriesof tensors of E, denoted by T ((E)) is defined by the following:

T ((E)) := a = (an)n∈N|∀n ∈ N : an ∈ E⊗n

T ((E)) has an algebra structure when endowed with the operations defined by the following:for a = (an)n∈N and b = (bn)n∈N in T ((E)) and λ ∈ R:

(Addition): a+ b = (an + bn)n∈N,

(Multiplication): a⊗ b =

(n∑

k=0

ak ⊗ bn−k

)

n∈N

,

(Multiplication by a scalar): λ.a = (λan)n∈N.

The invertible elements of the tensor algebra can be easily identified:

Proposition 2.9. [6] Let E be a real vector space. (T ((E)),+, .,⊗) is a non-commutative(assuming dim(E)≥ 2) unital algebra with unit 1 = (1, 0, 0, . . .). An element a = (an)n∈N inT ((E)) is invertible if and only if a0 6= 0. In this case, its inverse is given by the well-definedseries:

a−1 =1

a0

∑

n≥0

(1−

a

a0

)n

Notation 2.10.

T ((E)) = a = (an)n∈N ∈ T ((E))| a0 = 1

We now proceed to the definition of the signature of a path. This is given by the sequenceof its iterated integrals and is a well studied subject since [9]:

Definition 2.11. Let E be a Banach space and T ≥ 0. Let x : [0, T ] → E be a path ofbounded variation. For (s, t) ∈ ∆[0,T ], we define the following sequence by induction (well-defined by Stieltjes’ integration theory):

S0(x)(s,t) = 1Sn(x)(s,t) =

∫[s,t]

Sn−1(x)(s,u) ⊗ dxu , ∀n ∈ N∗

For every pair (s, t) ∈ ∆[0,T ], the sequence (Sn(x)(s,t))n∈N, simply denoted S(x)(s,t), is calledthe signature of x over [s, t]. For N ∈ N∗, (Sn(x)(s,t))n≤N , simply denoted SN(x)(s,t), is calledthe truncated signature of x over [s, t] of degree N .

The label “signature”, implicitely implying the full characterization of a path, can bejustified at many levels:

• Lyons and Hambly in [21] show that the signature of a path of bounded variationfully and uniquely characterizes the path in question up to a tree-like equivalence.This result has been extended later in more general contexts. Let us cite in particularthe work of Le Jan and Qian [22] for sample paths of the Brownian motion and thatof Boedihardjo, Geng, Lyons and Yang [4] for (weakly) geometric rough paths (thatwe will introduce later).


• In the context of differential equations, the signature of the control signal is the onlyneeded information to get the solution. This can be shown for example easily andexplicitely in the case where the vector fields in the differential equation are linearand continuous (details in [6] for example).

Lemma 2.12. [6] Let E be a vector space. Let m ∈ N be an integer and define

Bm = a = (an)n∈N|∀i ∈ 0, . . . , m ai = 0

Then is Bm an ideal of T ((E)).

Definition 2.13. [6] Let E be a vector space. Let m ≥ 0 be an integer. The truncatedtensor algebra of order m of E, denoted by T (m)(E), is the quotient algebra T ((E))/Bm. Wewill denote the canonical homomorphism T ((E)) → T ((E))/Bm by πm. There is a naturalidentification between T (m)(E) and

⊕0≤i≤m

E⊗i.

Definition 2.14 (Action of the Symmetric Group on Tensors). [28] Let n ∈ N∗, σ ∈ Sn andE be a vector space. We define the action of σ on the homogenous tensors of E of order nas a linear map by the following:

∀x1, x2, . . . , xn ∈ E σ(x1 ⊗ x2 ⊗ · · · ⊗ xn) = xσ(1) ⊗ xσ(2) ⊗ · · · ⊗ xσ(n)

Definition 2.15. Let E be a vector space. A norm ‖.‖ on T ((E)) is said to be admissible ifthe two following properties hold:

(1) ∀n ∈ N∗, ∀σ ∈ Sn, ∀x ∈ E⊗n ‖σx‖ = ‖x‖.(2) ∀n,m ∈ N∗, ∀x ∈ E⊗n, ∀y ∈ E⊗m ‖x⊗ y‖ ≤ ‖x‖‖y‖.

We will assume in the rest of this paper that (E⊗n)n≥1 are endowed with admissible norms.A discussion on certain basic properties of norms on tensor product spaces can be found forexample in [5].

2.1.3. Some algebraic and analytic properties of signatures. We resume the notations of theprevious subsection. Let S (respectively Sn, for n ∈ N∗) denote the map giving the signature(resp. the truncated signature of degree n) over [0, T ] of E-valued paths defined on [0, T ]that have bounded variation. We give below three basic properties of signatures that will bekey in defining rough paths in the next subsection. For a vector space E, we identify linearforms on T ((E)) with the elements of the vector space T (E∗) (the tensor polynomials of E∗,E∗ being the space of linear forms on E). We start by what we will be calling the shuffleproduct property:

Lemma 2.16. [6] Let E be a Banach space. For every two linear forms e and f defined onT ((E)), there exists a linear form denoted by e f such that:

∀x ∈ V1([0, T ], E) e(S(x)).f(S(x)) = (e f)(S(x))

e f is called the shuffle product of e and f .

Definition 2.17. Let t ∈ [0, T ] and x : [0, t] → E and y : [t, T ] → E be any two paths. Wecall the concatenation of x and y the path denoted x ∗ y defined over [0, T ] by:

(x ∗ y)u = xu , if u ∈ [0, t](x ∗ y)u = yu − yt + xt , if u ∈ [t, T ]

We define in a similar way the concatenation of any two paths defined on two adjacentsegments.


Remark 2.18. The concatenation operation is associative.

The second property that is of interest to us is what we call the multiplicativity property.The following theorem is due to Chen [10]. For a proof, see also [6]:

Theorem 2.19. Let t ∈ [0, T ]. Consider x ∈ V1([0, t], E) and y ∈ V1([t, T ], E). Thenx ∗ y ∈ V1([0, T ], E) and:

S(x ∗ y)(0,T ) = S(x)(0,t) ⊗ S(y)(t,T )

Remark 2.20. Theorem 2.19 is usually used in the following way (which is equivalent tothe previous formulation): for every path x ∈ V1([0, T ], E) and s, u, t ∈ [0, T ] such thats ≤ u ≤ t, we have:

S(x)(s,t) = S(x)(s,u) ⊗ S(x)(u,t)

Remark 2.21. For x ∈ V1([0, T ], E) and for (s, t) ∈ ∆[0,T ], S(x)(s,t) ∈ T ((E)) and istherefore invertible (proposition 2.9). Consequently, using theorem 2.19:

S(x)(s,t) = (S(x)(0,s))−1 ⊗ S(x)(0,t).

It is then equivalent to either study the path u 7→ S(x)(0,u) on the interval [0, T ] or (s, t) 7→S(x)(s,t) on the simplex ∆[0,T ].

Finally, the norms of the successive terms in a signature of a path of bounded variationdecay in a factorial way with repect to the 1-variation:

Theorem 2.22. [24] Let x be in V1([0, T ], E), then:

∀n ∈ N∗ ∀(s, t) ∈ ∆[0,T ] ‖Sn(x)(s,t)‖ ≤‖x‖n1,[s,t]n!

.

2.2. Rough Paths. The theory of rough paths generalizes the concept of signatures to moreirregular paths and provides the tools to solving differential equations driven by these withouthaving to build a whole new theory of integration for each one of them (as in Ito’s calculus).The concept of rough paths finds its source in the signature and the main analytic andalgebraic properties that it satisfies. The space of geometric rough paths is indeed simplydefined as the completion of that of signatures of paths with bounded variation under asuitably chosen metric similar to the p-variation metric for paths introduced in section 2.1.1.We introduce here the basic definitions and constructions.

2.2.1. Multiplicative functionals. The appropriate higher order generalisation of the linearityof the integral for iterated integrals and of Chen’s multiplicativity property for signatures isexpressed in terms of multiplicative functionals:

Definition 2.23. [6, 25] Let E be a normed vector space and T ≥ 0. Let X be a mapon ∆[0,T ] with values in T ((E)) (respectively in T (n)(E), with n ∈ N∗). X is said to be amultiplicative functional (resp. a multiplicative functional of degree n) if the following holds:

(1) X is continuous.(2) ∀t ∈ [0, T ] X(t,t) = 1.(3) X is multiplicative: ∀ 0 ≤ s ≤ u ≤ t ≤ T X(s,t) = X(s,u) ⊗X(u,t).

Remark 2.24. When no confusion is possible, we may use the term multiplicative functionalwith no reference to its degree being finite or not.

Remark 2.25. We will use the notation X i for the component of X of degree i.


It is clear, by Chen’s theorem 2.19, that, for every path x ∈ V1([0, T ], E), S(x) is amultiplicative functional and that for every n ∈ N∗, Sn(X) is a multiplicative functional ofdegree n.

2.2.2. p-variation metric and rough paths. We generalise now the notion of p-variation:

Definition 2.26. [6, 25] Let E be a normed vector space and T ≥ 0. Let p ≥ 1. Let Xbe a map on ∆[0,T ] with values in T ((E)) (respectively in T (n)(E), with n ∈ N∗) and ω be acontrol over [0, T ]. X is said to have a finite p-variation over [0, T ] controlled by ω if:

∀i ∈ N∗(resp.∀i ∈ [[1, n]]), ∀ 0 ≤ s ≤ t ≤ T : ‖X i(s,t)‖ ≤

ω(s, t)ip

βp

(ip

)!

where we write x! for Γ(x+1), with Γ being the usual extension of the factorial (the Gammafunction) and:

βp = p

(1 +

∞∑

k=1

(2

k

) [p]+1p

)

If there exists a control such that the previous properties holds, we may say that X has a finitep-variation over [0, T ] without mentioning the control. We denote by C0,p(∆[0,T ], T

[p](E)) the

set of continuous paths defined over ∆[0,T ] with values in T [p](E) and that have finite p-variation.

We see, that for n = 1, the concept of p-variation in definition 2.26 is the same as theone introduced in subsection 2.1.1. By theorem 2.22, signatures have finite 1-variation. Thep-variation control substitutes then the factorial decay property of signatures.

Remark 2.27. One can also easily note that for 1 ≤ q ≤ p, a multiplicative functional offinite q-variation is of finite p-variation.

Lyons’ extension theorem states that a multiplicative functional of finite p-variation isuniquely determined by its terms of degree less than or equal to [p]:

Theorem 2.28 (Extension theorem). [6, 25] Let p ≥ 1 and n ∈ N∗∪∞ such that n ≥ [p].Let E be a Banach space. Let X be a multiplicative functional of degree [p] in E that hasa finite p-variation over [0, T ] controlled by a control function ω. There exists a unique

multiplicative functional X of degree n that has a finite p-variation over [0, T ] and such thatπ[p](X) = π[p](X). Furthermore, the p-variation of X is also controlled by ω.

Remark 2.29. Theorem 2.28 partially encompasses Young’s integration theory [30] in thesense that it allows the construction of the signature of a path of finite p-variation, whenp < 2, using only the increments of the path.

The signature of a path of bounded variation is, therefore, by theorem 2.28, the onlymultiplicative functional with finite 1-variation which component of degree 1 corresponds tothe increments of said path.

We are now ready to give a definition for rough paths:

Definition 2.30 (Rough Paths). [6, 25] Let p ≥ 1 and E be a Banach space. A p-rough pathis a multiplicative functional of degree [p] that has finite p-variation. The set of all p-roughpaths over [0, T ] with values in E will be denoted by Ωp([0, T ];E), while that of p-rough paths

over sub-segments of [0, T ] will be denoted by Ω[0,T ]p (E).


Definition 2.31. We define on C0,p(∆[0,T ], T[p](E)) the p-variation metric, denoted by dp,

by the following:

dp(X, Y ) = max0≤i≤[p]

supD∈D[0,T ]

(∑

D

‖X i(tj ,tj+1)

− Y i(tj ,tj+1)

‖p

i

) 1p

for all X, Y ∈ C0,p(∆[0,T ], T[p](E)).

(For a subdivision D = (tj)0≤j≤n, we write∑D

a(tj ,tj+1) :=n−1∑j=0

a(tj ,tj+1).)

Let us note the following important property:

Theorem 2.32. [26] (Ωp([0, T ];E), dp) is a complete metric space.

2.2.3. Geometric p-rough paths. We introduce now geometric rough paths, a central notionin our paper. For more details, see [6] or [25] (or [15] for an extensive study of the subjectin the finite-dimensional case motivated with examples from stochastic analysis).

Definition 2.33. Let T ≥ 0 and E be a real Banach space. We define the set of E-valuedgeometric p-rough paths over [0, T ] to be the closure of the set S[p](x)|x ∈ V1([0, T ], E)in the p-variation metric. It is denoted by GΩp([0, T ];E). The set of E-valued geometric

p-rough paths over sub-segments of [0, T ] will be denoted GΩ[0,T ]p (E).

Rough paths do not take into account the shuffle product property satisfied by signatures.Geometric rough paths do not have this “setback” by continuity of linear forms:

Proposition 2.34. Let E be a Banach space and T ≥ 0. Let X be a geometric p-rough

path over [0, T ] in E. Then X is an element of Ω[0,T ]p (E) (in particular, X is multiplicative

and has finite p-variation) and for every (s, t) ∈ ∆[0,T ], X(s,t) satisfies the shuffle productproperty.

Remark 2.35. Let (G[p],⊗) be the Carnot group of the elements in the tensor algebra ofdegree at most [p] satisfying the shuffle product property, endowed with the homogeneousnorm: ‖.‖ : (1, g1, . . . , g[p]) 7→ max1≤i≤[p] ‖i!gi‖

1/i. It is interesting to see the analogy between

the restriction of dp to the space of geometric rough paths and the p-variation norm for pathstaking their values in (G[p],⊗) (see [27] for details).

We can also define the concatenation of paths taking their values in the truncated tensoralgebra:

Definition 2.36. Let n ∈ N∗. Let E be a vector space. Let s, u, t ∈ R such that s ≤ u ≤ t.Let X (resp. Y ) be a functional defined on ∆[s,u] (resp. on ∆[u,t]) with values in T n(E). Wedefine the concatenation of X and Y , denoted X ∗ Y , to be the functional over ∆[s,t] definedas follows: for (a, b) ∈ ∆[s,t]

(X ∗ Y )(a,b) =

X(a,b) , if b ≤ uX(a,u) ⊗ Y(a,u) , if a ≤ u ≤ bY(a,b) , if u ≤ a

The following theorem is straight-forward:


Theorem 2.37. Let p ≥ 1. Let E be a Banach space. Let s, u, t ∈ R such that s ≤ u ≤ t.Let X (resp. Y ) be a functional defined on ∆[s,u] (resp. on ∆[u,t]) with values in T [p](E).Then:

• If X and Y are multiplicative functionals, then X ∗ Y is a multiplicative functional;• If X and Y have finite p-variation, then X ∗ Y has finite p-variation;• If X and Y are geometric p-rough paths, then X ∗ Y is a geometric p-rough path.

It will be useful to us to attach a starting point to our geometric rough paths as we will bemostly dealing with integrals of rough paths and rough paths on manifolds; both of whichrequire a starting point. In the next few sections, a geometric rough path in a Banach spaceE will be a pair (x,X), where X is a geometric rough path in the sense of definition 2.33and x ∈ E is called the starting point. We will identify the space of geometric rough paths

with starting points with the Cartesian product E ×GΩ[0,T ]p (E).

On E × GΩ[0,T ]p (E) we define a metric dp as the product metric of dp and the norm on E,

i.e.:

dp((x,X), (y, Y )) = max(‖x− y‖, dp(X, Y ))

In other situations, it will be more convenient to attach a trace to a rough path instead ofa starting point (i.e. a path which increments correspond the element of degree 1 in saidrough path).

Definition 2.38. Let E be a Banach space and p ≥ 1. For an open subset U of E, a localgeometric p-rough path in U is a triple (x,X, J) such that:

• J is a compact interval.• x is a U-valued path over J .• X is an E-valued geometric p-rough path over J which trace is x, i.e. for all (s, t) ∈∆J , X

1s,t = xt − xs.

The set of local geometric p-rough paths in U defined over compact sub-intervals of I willbe denoted GΩI

p(U ;E), that of local geometric p-rough paths in U defined over a compactinterval J will be denoted GΩp(J ;U ;E).

Remark 2.39. The trace of a geometric p-rough path is trivially a path of finite p-variation.

Remark 2.40. It goes without saying that rough paths with starting points easily identifywith local rough paths. Indeed, if J is of the form, say, [0, T ], we will identify the local roughpath (x,X) (omitting the interval J when no confusion is possible) with the rough path withstarting point (x(0), X).

2.2.4. The integral of Lipschitz one-forms along geometric p-rough paths. When trying tomake sense of integrals of one-forms along rough paths, it is very important (for examplein regard to solving a differential equation) to be able to control the smoothness (in termsof variation) of the image of the path under the one-form. It appears that Lipschitz maps,in the sense of Stein [29], are the appropriate type of maps to use in this context (for anextensive dedicated study of these maps, see for example [5]):

Definition 2.41. Let n ∈ N and 0 < ε ≤ 1. Let E and F be two normed vector spaces and Ube a subset of E. Let f 0 : U → F be a map and for every k ∈ [[1, n]], let fk : U → Ls(E

⊗k, F )be a map with values in the space of the symmetric k-linear mappings from E to F .


For k ∈ [[0, n]], the map Rk : E × E → L(E⊗k, F ) defined by:

∀x, y ∈ U, ∀v ∈ E⊗k : fk(x)(v) =n∑

j=k

f j(y)(v ⊗ (x− y)⊗(j−k)

(j − k)!) +Rk(x, y)(v)

is called the remainder of order k associated to f = (f 0, f 1, . . . , fn).The collection f = (f 0, f 1, . . . , fn) is said to be Lipschitz of degree n + ε on U (or in shorta Lip− (n + ε) map) if there exists a constant M such that for all k ∈ [[0, n]], x, y ∈ U andv1, . . . , vk ∈ E:

(1) ‖fk(x)(v1 ⊗ · · · ⊗ vk)‖ ≤M‖v1 ⊗ · · · ⊗ vk‖;(2) ‖Rk(x, y)(v1 ⊗ · · · ⊗ vk)‖ ≤M‖x− y‖n+ε−k‖v1 ⊗ · · · ⊗ vk‖.

The smallest constant M for which the properties above hold is called the Lip− (n+ ε)-normof f and is denoted by ‖f‖Lip−(n+ε). The set of all Lip − (n + ε) maps defined on U withvalues in F will be denoted Lip(n + ε, U, F ).

Remark 2.42. Let us stress that the above definition is purely quantitative and makes senseeven on discrete sets. On any non-empty open subset of U , f 1, . . . , fn are the successivederivatives of f 0. However, these maps are not necessarily uniquely determined by f 0 onan arbitrary set U . Keeping this in mind, if f 0 : U → F is a map such that there existf 1, . . . , fn such that (f 0, f 1, . . . , fn) is Lip− (n+ε), we will often say that f 0 is Lip− (n+ε)with no mention of f 1, . . . , fn.

Theorem 2.43. [6, 25] Let γ, p ∈ R such that γ > p ≥ 1. Let E and F be two Banachspaces. Let α : E → Lc(E, F ) be a Lip− (γ−1) one-form. There exists a unique continuousmap:

Iα : (E ×GΩ[0,T ]p (E), dp) −→ (GΩ[0,T ]

p (F ), dp)

that extends Young’s integration theory, i.e. for all x ∈ V1([0, T ], E), Iα(x0, S[p](x)) =S[p](

∫α(x)dx). For a geometric p-rough path with a starting point (x0, X) or its correspond-

ing local rough path (x,X), we denote:

Iα(x0, X) =

∫α(x0, X)dX =

∫α(x,X)dX

3. The Lipschitz geometry

In this section, we review two of the findings of [8] that are most relevant to our work inthis paper: the Lipschitz structure and geometric rough paths on manifolds.

3.1. Lipschitz structures. We first introduce the general definitions of Lipschitz manifoldsand Lipschitz maps and one-forms on them.

Definition 3.1. [8] Let γ > 0. Let n ∈ N∗ and let M be a n-topological manifold. Let Ibe a countable set and, for every i ∈ I, Ui be an open subset of M and φi : M → Rn be acompactly supported map such that its restriction on Ui defines a homeomorphism. We saythat ((φi, Ui))i∈I is a Lipschitz-γ atlas if the following properties are satisfied:

• (Ui)i∈I is a pre-compact locally finite cover of M ;• For every i ∈ I: φi(Ui) = B(0, 1).• There exists δ ∈ (0, 1), such that (U δ

i )i∈I covers M , where, for every i ∈ I: U δi =

φ−1i |Ui

(B(0, 1− δ));


• There exists L > 0, such that, for every i, j ∈ I, φj (φi|Ui)−1 : B(0, 1) → Rn is

Lipschitz-γ and ‖φj (φi|Ui)−1‖Lip−γ ≤ L.

With the constants above, we will say that M is a Lipschitz-γ manifold with constants (δ, L).

Example 3.2. As one would expect, finite-dimensional vector spaces can be endowed with anatural Lipschitz-γ manifold structure of any degree γ ≥ 1.

Proof. Indeed, let γ ≥ 1. Let V be a finite dimensional space and let (e1, . . . , en) be a basisfor V . Let ϕ be a Lip-γ extension on V of IdB(0,1) with support in B(0, 2). For x ∈ V , letϕx be the map: y 7→ ϕ(y − x). Then (ϕx, B(x, 1))x∈I is a Lip-γ atlas on V , where:

I =

n∑

i=1

ki2ei| k1, . . . , kn ∈ Z

Example 3.3. [8]1 Let n ∈ N∗ and 0 < ε ≤ 1. A Cn+1 structure on a compact manifoldinduces a natural Lip-(n+ ε) structure.

Definition 3.4. Let γ0 ≥ 1 and d ∈ N∗. Let M be a d-dimensional Lip− γ0 manifold withan atlas (φi, Ui), i ∈ I and E be a normed vector space.

• A map f :M → E is said to be Lip− γ, for γ ≤ γ0, if there exists a constant C suchthat, for every i ∈ I, f φi

−1|Ui

: B(0, 1) → E is Lip− γ with a Lipschitz norm at mostC. The smallest constant C for which this property holds is called the Lip− γ normof f and is denoted by ‖f‖Lip−γ.

• An E-valued one-form α on M is said to be Lip− γ, for γ ≤ γ0 − 1, if there exists aconstant C such that, for every i ∈ I:

(φi−1|Ui)∗α : B(0, 1) → L(Rd, E)

is Lip− γ with a Lipschitz norm at most C. The smallest constant C for which thisproperty holds is called the Lip− γ norm of α and is denoted by ‖α‖Lip−γ.

Remark 3.5. An important property that is omitted in [8] and that we underline in theabove definition is that, on a Lip − γ0 manifold, it does not make sense geometrically todefine Lip − γ maps or Lip − (γ − 1) one-forms if γ > γ0. Indeed, with the notations ofdefinition 3.4, if for i ∈ I, f φi

−1|Ui

: B(0, 1) → E is Lip − γ (where γ > γ0), then, basedonly on the definition of a Lip− γ0 atlas, one cannot show that, for j ∈ I, the restriction off φj

−1|Uj∩Ui

: B(0, 1) → E on any non-trivial subset of Uj ∩ Ui is smoother than Lip − γ0.

The same principle applies to defining Lipschitz one-forms. A more rigorous way to statethe above is obtained by translating it in the language of equivalent Lipschitz structures (see[8] for a definition).

3.2. Rough paths on a manifold. In this subsection, we generalize the notion of roughpaths to manifolds (as presented in [8]). As we don’t have a natural notion of linearityand iterated integrals on a manifold, we have to consider a different approach than the onearising from the p-variational properties of paths and signatures. As we will hint to later,integrals of Lipschitz one-forms along rough paths do characterize the path; this is the firstdirection that we will take to define our rough paths. Additionally, in the absence of anatural translation, a rough path on a manifold comes attached with a starting point.

1This is a slightly stronger result than in [8] but its proof is practically the same.


Definition 3.6. [8] Let γ0, p ∈ R such that γ0 > p ≥ 1 and T ≥ 0. Let M be a Lip − γ0manifold and x ∈ M . X is a geometric p-rough path over [0, T ] on M starting at x if, forevery γ ∈ R such that γ0 ≥ γ > p and every Banach space E, the following conditions aresatisfied:

(1) X maps Lip− (γ−1) E-valued one-forms on M to E-valued geometric p-rough paths(in the classical sense).

(2) There exists a control ω such that for every E-valued Lip − (γ − 1) one-form α onM , X(α) is controlled by ‖α‖Lip−(γ−1)ω, i.e.:

∀(s, t) ∈ ∆[0,T ], ∀i ∈ [[1, [p]]] : ‖X(α)i(s,t)‖ ≤(‖α‖Lip−(γ−1)ω(s, t))

i/p

βp(i/p)!

(3) (Chain rule) For every Banach space F and every compactly supported Lip− γ mapψ :M → F and every E-valued Lip− (γ − 1) one-form α on F we have:

X(ψ∗α) =

∫α(ψ(x),X(ψ∗))dX(ψ∗)

Contrary to the classical framework, in the context of manifolds, we do not need to makea difference between rough paths and geometric rough paths (as only the latter are defined).Consequently, we drop the word “geometric” when talking about geometric rough paths onmanifolds. Moreover, in the classical sense, geometric rough paths are uniquely determinedby the values of the integrals of compactly supported one-forms along them. In order tomake the correspondance one-to-one between the concepts of classical geometric rough paths(on finite-dimensional spaces) and rough paths on the same space when endowed with itscanonical Lipschitz structure, we define the following equivalence relation:

Definition 3.7. [8] Let γ0, p ∈ R such that γ0 > p ≥ 1. Let M be a Lip − γ0 manifold.

We say that two p-rough paths X and X on M are equivalent, and we write X ∼ X, ifthey have the same starting point and if, for every γ ∈ R such that γ0 ≥ γ > p and forevery Lip − (γ − 1) compactly supported Banach space valued one-form α on M , we have

X(α) = X(α).

We give now a hint on how to build a one-to-one correspondance between rough paths inthe classical sense and in a Lip-γ manifold, when said manifold is a finite-dimensional vectorspace V (see [8] for the complete proof):

• Given a geometric p-rough path (x,X) on V , we define the rough path X in the man-ifold by the starting point x and the functional sending every compactly supportedBanach space-valued Lip-(γ − 1) one form α to the classical rough path:

X(α) =

∫α(x,X)dX

• Conversely, given a rough path X on V in the manifold sense, we are tempted toretrieve a rough path in the classical sense by integrating dIdV along X. Since dIdVis not compactly supported (this is important for the definition to be consistent acrossthe equivalence classes of the equivalence relation introduced in definition 3.7) andthat IdV is not Lip-γ, we can intuitively replace IdV with a compactly supportedLip-γ extension of its restriction on a set containing the “support” of X. The notionof support is not yet defined at this point but we can still have a good guess at a setcontaining it by using the control of the p-variation of X.


When working on a manifold, one has to ensure that certain properties are invariant by thechange of charts (or, in other words, by local reparametrisation). For this reason, we definethe pushforward of rough paths by conveniently chosen maps:

Lemma 3.8. [8] Let γ0, p ∈ R such that γ0 > p ≥ 1. Let M and N be Lip − γ0 manifoldsand f : M → N be a map such that there exists a constant Cf such that, for all γ ∈ (p, γ0]and every Lip− (γ − 1) Banach space valued one-form α on M , we have:

‖f ∗α‖Lip−(γ−1) ≤ Cf‖α‖Lip−(γ−1)

Then f induces a pushforward map f∗ from p-rough paths on M to p-rough paths on Ndefined as follows: for every p-rough path X over [0, T ] on M starting at x, f∗X starts atf(x) and for every Lip− (γ − 1) Banach space valued one form α on M , where γ ∈ (p, γ0],f∗X(α) is given by f∗X(α) = X(f ∗α).

The next proposition shows that there exists a particular class of Lipschitz maps thatinduce pushforwards of rough paths. In particular, we can ascertain that the pushforwardsof rough paths by coordinate maps or transition maps are also rough paths:

Proposition 3.9. [8]2 Let γ0 ≥ γ > 1 and let M be a Lip-γ0 manifold and W be a normedvector space. Let f :M →W be a Lip−γ map and α be a Lip− (γ−1) Banach space valuedone-form on W (or defined on a subset of W containing f(M)). Then f ∗α is Lip-(γ − 1)and there exists a constant Cγ depending only on γ such that:

‖f ∗α‖Lip−(γ−1) ≤ Cγ‖α‖Lip−(γ−1)‖f‖Lip−γ max(‖f‖γ−1Lip−γ , 1)

Like in the classical case, we can define the concatenation of two rough paths. In the con-text of manifolds, this is important on its own since one usually works locally on coordinatedomains to solve ordinary or rough differential equations before attempting to make senseof a global solution via a concatenation procedure:

Definition 3.10. [8] Let γ0 > p ≥ 1 and s ≤ u ≤ t. Let M be a Lip− γ0 manifold. Let X(respectively Y) be a p-rough path onM over [s, u] (resp. [u, t]). We define the concatenationof X and Y, denoted by X ∗ Y, to be the functional over [s, t] mapping every Banach space-valued Lip-(γ − 1) compactly supported one-form α (for every γ0 ≥ γ > p) to the classicalrough path X(α) ∗ Y(α).

Unlike the classical case, the concatenation of two rough paths on a manifold is notnecessarily a rough path. This is due to the fact that rough paths on a manifold comeattached with a starting point and that we have no natural notion of translation. Therefore,for this concatenation to be a rough path, we have to make sure that the two rough pathsin question have starting and “ending” points that agree in the following sense:

Definition 3.11. Let γ0 > p ≥ 1 and s ≤ t. Let M be a Lip − γ0 manifold. Let X be ap-rough path on M over [s, t] with starting point x and let y ∈ M . We say that X has anend point consistent with y if for every Banach space-valued compactly supported Lip-γ mapf on M (for every γ0 ≥ γ > p), we have:

f(x) + X(f∗)1s,t = f(y)

2Following [5], we correct the exponent in the inequality compared to the result that appeared in [8]and drop furthermore the dependence on the dimension of the manifold. Said inequality can also be madesharper using improved estimates in [5].


In this case, we can check the consistency condition for the concatenation of two roughpaths and prove that it is also a rough path:

Proposition 3.12. [8] Let γ0 > p ≥ 1 and s ≤ u ≤ t. Let M be a Lip− γ0 manifold. Let X(respectively Y) be a p-rough path on M over [s, u] (resp. [u, t]) with starting point x (resp.y). We assume that X has an end point consistent with the starting point of Y. Then X ∗Ydefines a p-rough path on M over [s, t] with x as a starting point.

Well-defined concatenations of rough paths constitute an associative operation:

Lemma 3.13. [8] Let γ0 > p ≥ 1 and M be a Lip−γ0 manifold. Let X, Y and Z be p-roughpaths on M such that:

• X has an end point consistent with the starting point of Y;• Y has an end point consistent with the starting point of Z.

Then:

• X has an end point consistent with the starting point of Y ∗ Z;• X ∗ Y has an end point consistent with the starting point of Z;• X ∗ (Y ∗ Z) = (X ∗ Y) ∗ Z

In this case, we denote X ∗ Y ∗ Z := X ∗ (Y ∗ Z).

Since this notion of rough paths does not attach, for the moment, an underlying path onthe manifold to a rough path, we define a notion of support based on the images of one-formssupported in all possible open sets:

Definition 3.14. [8] Let γ0 > p ≥ 1 and M be a Lip−γ0 manifold. Let X be a p-rough pathon M with starting point x.

(1) For an open subset U of M , we say that X misses U if, for every γ0 ≥ γ > p andevery Banach space-valued compactly supported Lip-(γ − 1) one-form α on M withsupport in U , we have X(α) = 0.

(2) We define the support of the rough path X as the closed set:

supp(X) := x⋃(M −

⋃

X misses U

U

)

Naturally, this notion of support is consistent with the definition of support for a classicalrough path:

Proposition 3.15. [8] Let γ0, p ∈ R such that γ0 > p ≥ 1 and T > 0. Let V be a finitedimensional vector space endowed with its canonical structure of a Lip-γ0 manifold. Let(x,X) be a geometric p-rough path on V over [0, T ] (in the classical sense), and let X bethe p-rough path associated to it in the manifold sense, i.e. for every Banach space-valuedLip− (γ − 1) compactly supported one-form α on V : X(α) =

∫α(x,X)dX. Then:

supp(X) =x+X1

0,t, t ∈ [0, T ]

Like in the classical case, the support of a rough path on a manifold can be shown to becompact:

Theorem 3.16. [8] Let γ0 > p ≥ 1. Let M be a Lip− γ0 manifold and X be a p-rough pathon M . Then the support of X is compact.


The claim of the previous theorem can be shown by writing a rough path on a manifoldas the concatenation of rough paths that have their support contained in the domain of onlyone chart at a time as described in the following theorem. This paper will generalise thisdecomposition of rough paths on manifolds into a collection of classical “local” rough pathsthat are compatible in a certain sense.

Theorem 3.17. [8] Let γ0 > p ≥ 1 and T ≥ 0. Let M be a Lip − γ0 manifold. LetX be a p-rough path on M defined over an interval [0, T ]. Then there exists a subdivisionD = (si)0≤i≤N of [0, T ] and a collection (Xi, xi, (φi, Ui))1≤i≤N such that, for every i ∈ [[1, N ]]:

(1) Xi is a p-rough path on M defined over [si−1, si] with starting point xi.(2) (if i < N) xi+1 is consistent with the endpoint of Xi.(3) (φi, Ui) is a Lip-γ0 chart on M such that

supp(Xi) ⊆ Ui

and such that for every Banach space-valued Lip-(γ − 1) one-form α defined on M(where γ0 ≥ γ > p) and compactly supported Lip-(γ − 1) one-form ξα defined on Rd

which agrees with (φi−1|Ui)∗(α) on B(0, 1), we have:

Xi(α) = ((φi)∗X)[si−1,si](ξα)

(4) X = X1 ∗ · · · ∗ XN .

(Xi, xi, (φi, Ui))1≤i≤N is called a localising sequence for X.

4. Local rough paths

4.1. Intervals. We briefly present here some basic topological properties of covers of inter-vals which will be of use when studying rough paths locally.

Definition 4.1. Let M be a topological space. Let J be a subset of M and (Ki)i∈I be acollection of subsets of M .

(1) (Ki)i∈I is said to be locally finite if for each x ∈ M there exists a neighbourhood Ux

of x such that at most finitely many Ki’s have non-empty intersection with Ux.(2) (Ki)i∈I is said to cover J (or is a cover of J) if J ⊆ ∪i∈IKi.(3) (Ki)i∈I is called a compact cover of J if it is locally finite, covers J and for i ∈ I, Ki

is compact subset of J .

The proof of the following lemma is straightforward:

Lemma 4.2. Each interval J of R admits a compact cover.

Lemma 4.3. Let M be a topological space, J be a compact subset of M and (Ki)i∈I be alocally finite collection of compact sets that cover J such that all the Ki’s intersect J . ThenI is finite.

Proof. For every x ∈ J , let Ux be an open neighbourhood of x that intersects only finitelymany of the Ki’s. Then (Ux)x∈J is an open cover of J . As J is compact, there exists a finitesubset J0 of J such that J0 = Ux; x ∈ J0 covers J . As every Ki intersects J , and hencean element in the finite set J0, and since every element of J0 intersects only finitely manyof the Ki’s, I must be finite.

Corollary 4.4. Let J be an interval and (Ki)i∈I be a compact cover for J . Then I iscountable. Moreover, if J is compact, then I is finite.


Proof. Let (Jn)n∈N be a non-decreasing sequence of compact intervals (in the sense of inclu-sion) such that J = ∪nJn. Let n ∈ N and define:

In = i ∈ I; Ki ∩ Jn 6= ∅

Then (Ki)i∈In is a locally finite collection of compact sets that cover Jn such that all the Ki’sintersect Jn. By lemma 4.3, In is finite. Note now that (In)n∈N is a non-decreasing sequenceof finite subsets of I. Moreover, I = ∪nIn. Hence, I is countable.The case when J is compact is covered by lemma 4.3.

Lemma 4.5. If (Ki)i∈I is a compact cover of an interval J , and if for each i ∈ I, (Kj)j∈Iiis a compact cover of Ki then Kj|j ∈ Ii, i ∈ I is also a compact cover of J .

Proof. First note that:

∪i∈I,j∈IiKj = ∪i∈I(∪j∈IiKj) = ∪i∈IKi = J

The Kj ’s (j ∈ ∪IIi) are all compact subsets of J . Let x ∈ J . Let Vx,J be a neighbourhoodof x in J that intersects finitely many Ki’s (i ∈ I). As Ii is finite for every i ∈ I (corollary4.4), then Vx,J intersects finitely many Kj’s (j ∈ ∪IIi).

Definition 4.6. Let (Ui)i∈I and (Vj)j∈J be two collections of sets. We say that (Vj)j∈J is arefinement of (Ui)i∈I if, for every j ∈ J , there exists i ∈ I such that Vj ⊆ Ui.

Lemma 4.7. Let O = (Oi)i∈I be a cover of an interval J by open sets and K = (Kh)h∈H bea compact cover for J . Then there exists a compact cover for J that is a refinement of bothO and K.

Proof. Let h ∈ H and x ∈ Kh. As x ∈ J , then there exists ix ∈ I and αx > 0 such that(x−αx, x+αx)∩J ⊆ Oix . As Kh is compact and ((x−αx/2, x+αx/2))x∈Kh

covers Kh, thenthere exists a finite subset P ⊆ Kh such that ([x − αx/2, x + αx/2])x∈P covers Kh. DefineIx = [x−αx/2, x+αx/2]∩Kh for every x ∈ P . Then, for every x ∈ P , Ix is a compact subsetof Kh that is contained in Oix. (Ix)x∈P is therefore a compact cover of Kh. We concludeusing lemma 4.5.

Proposition 4.8. Given any cover of a compact interval J by open sets (Oi)i∈I, there existsa (finite) subdivision (aj)0≤j≤n of J such that:

∀j ∈ [[0, n− 1]], ∃i ∈ I such that: [aj , aj+1] ⊆ Oi

Proof. By Lebesgue’s number lemma, let δ > 0 such that for every subset A of J of diameterless than δ there exist i ∈ I, such that A ⊆ Oi. It suffices to take any subdivision of J ofmesh less than δ to conclude the proof.

Proposition 4.9. Given any cover of a compact interval J by compact intervals (Ki)i∈I ,there exists a (finite) subdivision (aj)0≤j≤n of J such that:

∀j ∈ [[0, n− 1]], ∃i ∈ I such that: [aj, aj+1] ⊆ Ki

Proof. Without loss of generality, we may assume that J = [0, 1]. For every x ∈ [0, 1), weclaim that:

∃εx > 0, ∃i ∈ I : [x, x+ εx] ⊂ Ki

Indeed, assume the converse is true. Then there exists x ∈ [0, 1) such that:

∀n ∈ N∗, ∀i ∈ I, ∃xi,n ∈

[x, x+

1

n

]−Ki


Define I0 = i ∈ I, x ∈ Ki. Then necessarily I0 ( I. Indeed, if I0 = I, we let i0 be suchthat 1 ∈ Ki0 and then, by convexity of Ki0 , we have [x, 1] ⊂ Ki0, which contradicts ourassumption. Let i∗ ∈ I0. For all n ∈ N∗, let xn be an element of [x, x + 1

n] − Ki∗ . Then,

for all n ∈ N∗ and i ∈ I0, xn 6= x (since x ∈ Ki∗) and xn /∈ Ki (by the same convexityargument used above). Hence, (xn) is a sequence in the compact set ∪I−I0Ki converging tox, which leads to a contradiction. Therefore, the claim is true. For every i ∈ I, we writeKi = [si, ti]. Let i0 ∈ I such that Ki0 contains a neighbourhood of 0 with non-empty interiorand for every i ∈ I such that ti 6= 1, let ri ∈ I be such that there exists εi > 0 such that[ti, ti+ εi] ⊂ Kri. We define a0 = 0 and a1 = ti0 . Then a0 < a1 and [a0, a1] ⊂ Ki0 . We definethe rest of the subdivision in a recursive way: for q ≥ 1, given aq = tq∗ , if aq = 1, then weare done, otherwise we set aq+1 = trq∗ (we have then aq < aq+1 and [aq, aq+1] ⊂ Krq∗ ). Itis clear that this procedure converges in a finite number of steps and produces the desiredsubdivision.

4.2. Locally Lipschitz maps.

Definition 4.10. Let γ > 0. Let E and F be two normed vector spaces, U be a subset ofE and f : U → F be a map. We say that f is locally Lipschitz-γ if, for every x ∈ U , thereexists a neighborhood Vx,U of x in U such that f|Vx,U

is Lipschitz-γ. The set of all locallyLip− γ maps defined on U with values in F will be denoted Liploc(γ, U, F ).

Example 4.11. Lipschitz-γ maps are obviously locally Lipschitz-γ. Continuous linear andpolynomial maps are locally Lipschitz (of any degree).

Remark 4.12. One of the main reasons for introducing locally Lipschitz maps here insteadof working with Lipschitz maps that are classical in the setting of geometric rough paths is tobe able to use linear maps (such as the identity maps which are pivotal in the definition ofcategories) which are not Lipschitz in general3. Another possible solution to this issue is theuse of almost Lipschitz maps introduced in [5]. The results below generalise automatically tothis class of maps.

We recall here two important results on Lipschitz maps. They can be found for examplein [5] and [8]:

Theorem 4.13. 4 Let E, F and G be three normed vector spaces. Let U be a subset of Eand V be a subset of F . Let γ ≥ 1. We assume that (E⊗k)k≥1 and (F⊗k)k≥1 are endowedwith norms satisfying the projective property. Let f : U → F and g : V → G be two Lip− γmaps such that f(U) ⊆ V . Then g f is Lip− γ and there exists a constant Cγ (dependingonly on γ) such that:

‖g f‖Lip−γ ≤ Cγ‖g‖Lip−γ max(‖f‖γLip−γ, 1)

Lemma 4.14. Let n ∈ N, 0 < ε ≤ 1 and C ≥ 0. Let E and F be two normed vectorspaces and U be an open subset of E. Let f : U → F be a map and for every k ∈ [[1, n]], letfk : U → L(E⊗k, F ) be a map with values in the space of the symmetric k-linear mappingsfrom E to F . We consider the two following assertions:

(A1): (f, f 1, . . . , fn) is Lip− (n+ ε) and ‖f‖Lip−(n+ε) ≤ C.

3 a usual workaround in analysis is to use suitable Whitney extensions4see [5] for a slightly improved estimate


(A2): f is n times differentiable, with f 1, . . . , fn being its successive derivatives. ‖f‖∞,‖f 1‖∞, . . ., ‖fn‖∞ are upper-bounded by C and for all x, y ∈ U : ‖fn(x)− fn(y)‖ ≤C‖x− y‖ε.

Then (A1) ⇒ (A2). If furthermore U is convex then (A1) ⇔ (A2).

Using for example lemma 4.14, one can easily show the following result:

Proposition 4.15. Let n ∈ N∗. Let E and F be two normed vector spaces, U be an opensubset of E and f : U → F be a map of class Cn. Then f is locally Lip-n.

Unlike the notion of Lipschitzness, local Lipschitzness can easily be defined recursively onopen sets:

Lemma 4.16. Let γ > 1. Let E and F be two normed vector spaces, U be an open subset ofE and f : U → F be a differentiable map. Then its derivative df is locally Lipschitz-(γ− 1)if and only if f is locally Lipschitz-γ.

Proof. Notice that, on every ball B(x, α) ⊆ U on which df is Lipschitz-(γ − 1), one canuse for example the fundamental theorem of calculus to bound f and we deduce that therestriction of f on this set is Lipschitz-γ using lemma 4.14. The converse is obvious.

Following theorem 4.13, local Lipschitzness is conserved under composition:

Proposition 4.17. Let γ ∈ [1,+∞[. Let E, F and G be normed vector spaces, U be asubset of E and V be a subset of F . Let f : U → F and g : V → G be two locally Lipschitz-γmaps such that f(U) ⊆ V . Then g f is locally Lipschitz-γ.

Before we carry on, we need the following embedding theorem for which the completestatement and proof can be found for example in [5].

Theorem 4.18. Let γ, γ′ > 0 such that γ′ ≤ γ. Let E and F be two normed vector spacesand U be a subset of E. Let f : U → F be a Lip − γ map. Then f is Lip − γ′ and thereexists a constant Mγ,γ′ (depending only on γ and γ′) such that ‖f‖Lip−γ′ ≤Mγ,γ′‖f‖Lip−γ.

Locally Lipschitz maps conserve the smoothness (in the sense of variation) of paths:

Theorem 4.19. Let p, γ ∈ [1,+∞[. Let E and F be two normed vector spaces, U be asubset of E and J a compact interval. Let f : U → F be a locally Lip-γ map over U andx : J → U a path with finite p-variation. Then f x : J → F is of finite p-variation.

Proof. For every t ∈ J , let Bt be an open neighborhood of xt in U such that f|Btis Lipschitz-1

(following theorem 4.18); denote its Lip-1 norm by Mt. Then (x−1(Bt))t∈J is an open coverof J . Let (ai)0≤i≤n be a subdivision of J such that for all i ∈ [[0, n − 1]], there exists ti ∈ Jsuch that [ai, ai+1] ⊆ x−1(Bti) (proposition 4.8) and denote M = max1≤i≤nMti .

Let ω be a control of the p-variation of x over J . Let s, u ∈ J and let q, r ∈ [[0, n]] suchthat aq ≤ s ≤ · · · ≤ u ≤ ar. Then we have, from the fact that f is Lip-1 on each of the Bti ’s,


for i ∈ [[0, n− 1]], with a norm less than M :

‖f(xs)− f(xu)‖ ≤ ‖f(xs)− f(xaq+1)‖+r−2∑

k=q+1

‖f(xak)− f(xak+1)‖+

‖f(xar−1)− f(xu)‖

≤ M

(‖xs − xaq+1‖+

r−2∑k=q+1

‖xak − xak+1‖+ ‖xar−1 − xu‖

)

≤ M

(ω(s, aq+1)

1/p +r−2∑

k=q+1

ω(ak, ak+1)1/p + ω(ar−1, u)

1/p

)

which, using the super-additivity of ω and Jensen’s inequality, gives the control:

‖f(xs)− f(xu)‖p ≤Mpnp−1ω(s, u)

Therefore, f x is of finite p-variation (M and n do not depend on s or u).

4.3. A study of some properties of rough paths.

Lemma 4.20. Let E be a vector space and J be a compact interval. Let X and Y be twoE-valued multiplicative functionals on J . Let (Ki)i∈I be a compact cover for J by compactintervals such that, for all i ∈ I, X|Ki

and Y|Kiare equal. Then X and Y are equal on J .

Proof. Let (a, b) ∈ ∆J . Then (Ki∩ [a, b])i∈S, where S = i ∈ I|Ki∩ [a, b] 6= ∅, is a compactcover for [a, b] by compact intervals. Let (aj)0≤j≤n be a subdivision of [a, b] such that forall j ∈ [[0, n − 1]], there exists i ∈ S such that [aj , aj+1] ⊆ Ki (proposition 4.9). Then, byassumption: ∀j ∈ [[0, n− 1]] Xaj ,aj+1

= Yaj ,aj+1. Therefore:

Xa0,a1 ⊗ · · · ⊗Xan−1,an = Ya0,a1 ⊗ · · · ⊗ Yan−1,an

Which, by using the multiplicativity of X and Y , gives: Xa,b = Ya,b. Since this holds for all(a, b) ∈ ∆J , then X = Y .

Lemma 4.21. Let E be a vector space and J be a compact interval. Let (Ki)i∈I be a compactcover for J by compact intervals. Let (Xi)i∈I be a collection of E-valued multiplicative func-tionals such that, for all i ∈ I, Xi is defined over Ki and for i, j ∈ I such that Ki ∩Kj 6= ∅,we have Xi|Ki∩Kj

= Xj |Ki∩Kj. Then there exists a unique multiplicative functional X defined

over J such that, for all i ∈ I, X|Ki= Xi.

Proof. Let (aj)0≤j≤n be a subdivision of J and (ij)0≤j≤n be a finite sequence of elements of Isuch that for all j ∈ [[0, n−1]], we have [aj , aj+1] ⊆ Kij . Let (s, t) ∈ ∆J and let q, r ∈ [[1, n−1]]be such that:

aq−1 ≤ s ≤ aq ≤ · · · ≤ ar ≤ t ≤ ar+1

We set:Xs,t = Xiq−1(s, aq)⊗ · · · ⊗Xir(ar, t)

It is an easy exercise to check that X defines a multiplicative functional satisfying therequirements of the statements. The uniqueness of such functional is a consequence oflemma 4.20.

Before we make sure that the integral of a geometric rough path against a sufficientlysmooth locally Lipschitz one-form is indeed well defined, we need the following lemma quan-tifying the distance over an interval in the variation topology between two rough paths giventheir distance on a subdivision of that interval.


Lemma 4.22. Let E be a Banach space, p ∈ [1,+∞[ and T ≥ 0. Let Y and Z be twocontinuous maps from ∆[0,T ] to T [p](E) with finite p-variation. Given a subdivision D =(si)0≤i≤r of [0, T ], then we have:

dp(Y, Z) ≤ 31−1/p max1≤j≤[p]

‖(d[0,s1]p (Y, Z), · · · , d[sr−1,T ]p (Y, Z))‖lj

where, for a finite sequence x = (x1, x2, . . . , xn) and j > 0, we define:

‖x‖lj =

(n∑

i=1

|xi|j

)1/j

Proof. Let j ∈ [[1, [p]]]. For (s, u) ∈ ∆[0,T ], define Vs,u = Y js,u − Zj

s,u. Let ∆ = (ti)0≤i≤q be asubdivision of [0, T ]. Let i ∈ [[0, q − 1]]. Define the subdivision D ∩ [ti, ti+1] of [ti, ti+1] to be(mi

l) := (ti, sri, . . . , sri+ni, ti+1), where ri and ni are such that (if they exist) sri−1 < ti ≤ sri

and sri+ni≤ ti+1 < sri+ni+1. Then we have:

‖Vti,ti+1‖p/j ≤

(‖Vti,sri‖+

ri+ni∑l=ri

d[sl,sl+1]p (Y, Z)j + ‖Vsri+ni

,ti+1‖

)p/j

≤ 3p/j−1

(‖Vti,sri‖

p/j +

(ri+ni∑l=ri

d[sl,sl+1]p (Y, Z)j

)p/j

+ ‖Vsri+ni,ti+1

‖p/j

)

Using the inequality aα + bα ≤ (a + b)α, for a, b ≥ 0 and α ≥ 1 after summing over all i’s,we get:

∑

∆

‖Vti,ti+1‖p/j ≤ 3p/j−1

(r−1∑

i=0

d[si,si+1]p (Y, Z)j

)p/j

= 3p/j−1‖(d[0,s1]p (Y, Z), · · · , d[sr−1,T ]p (Y, Z))‖plj

which gives the result.

We can now show that the integral of a locally Lipschitz one-form along geometric roughpaths is, as expected, well defined and continuous when varying the path.

Theorem 4.23. Let p, γ ∈ [1,+∞[ such that γ > p and T ≥ 0. Let E and F be two Banachspaces and U be an open subset of E. There exists a unique map:

J : Liploc(γ − 1, U,L(E, F ))×GΩ[0,T ]p (U ;E) −→ GΩ[0,T ]

p (F )

such that if α : U → L(E, F ) is a Lip-(γ − 1) one-form and (x,X) is a local geometricp-rough path in U defined over an interval [s, t] ⊆ [0, T ] then:

J (α, (x,X)) =

∫α(x,X)dX

Moreover, for a locally Lip-(γ−1) one-form α : U → L(E, F ) and a subinterval [s, t] ⊆ [0, T ],the map:

Jα : (GΩp([s, t];U ;E), dp) −→ (GΩp([s, t];F ), dp)(x,X) 7→ J (α, (x,X))

is continuous.


Proof. Let α : U → L(E, F ) be a locally Lip-(γ−1) one-form and (x,X) ∈ GΩ[0,T ]p (U) defined

over a compact interval J . Let s, t ∈ ∆J . For u ∈ [s, t], let Ou be an open neighbourhoodof xu in U such that α|Ou

is Lip-(γ − 1). As (x−1(Ou))s≤u≤t is an open covering of [s, u], let(si)0≤i≤n be a subdivision of [s, t] such that for all i ∈ [[0, n− 1]], there exists ui ∈ [s, t] suchthat [si, si+1] ⊂ x−1(Oui

).Let i ∈ [[0, n − 1]]. The rough path (x,X)|[si,si+1] takes its values in Oui

on which α is Lip-(γ−1). Therefore, the geometric p-rough path

∫α(x(si), X)dX is well defined over [si, si+1].

By lemma 4.21, there exists a unique geometric p-rough path J (α, (x,X)) defined over [s, u]which agrees with it. This proves the existence of the map J . The uniqueness is shownfollowing the same reasoning and decomposition.Let (x,X) ∈ GΩp([0, T ];U ;E). For u ∈ [0, T ], let ηu > 0 be such that B(xu, ηu) ⊆ Uand α|B(xu,ηu) is Lip-(γ − 1). As before, let (si)0≤i≤n be a subdivision of [0, T ] such thatfor all i ∈ [[0, n − 1]], there exists ui ∈ [0, T ] such that x([si, si+1]) ⊂ B(xui

, ηui/2) and

denote η = mini ηui. Let ε > 0. Let β > 0 be such that for all i ∈ [[0, n − 1]] and

(z, Z) ∈ GΩp([si, si+1];B(xui, ηui

);E):

dp((x(si), X|[si,si+1]), (z(si), Z)) ≤ β ⇒ d[si,si+1]p

(∫α(x(si), X|[si,si+1])dX,

∫α(z(si), Z)dZ

)≤ ε

Let (y, Y ) ∈ GΩp([0, T ];U ;E) such that dp((x(0), X), (y(0), Y )) ≤ min(β, η)/4. For u ∈[0, T ], we have then:

‖x(u)− y(u)‖ ≤ ‖x(0)− y(0)‖+ ‖X10,u − Y 1

0,u‖ ≤ min(β, η)/2

Let i ∈ [[0, n− 1]]. For u ∈ [si, si+1], both xu and yu are in B(xui, ηui

), hence

d[si,si+1]p (Jα(x,X),Jα(y, Y )) ≤ ε

By lemma 4.22 , we deduce that:

dp(Jα(x,X),Jα(y, Y )) ≤ 31−1/prε

from which we deduce the continuity of Jα at (x,X).

Lemma 4.24. Let p, γ ∈ [1,+∞[ such that γ > p. Let E and F be two Banach spaces andU be an open subset of E. and J an interval. Let f : U → F be a locally Lip-γ map over Uand x : J → U a path with bounded variation. Then:

(f(x),

∫df(x, S[p](x))dS[p](x), J) = (f(x), S[p](f(x)), J)

Proof. First notice that S[p](f(x)) is well-defined since f(x) has bounded variation by theorem4.19. Let s, t ∈ ∆J :

(1) As df is continuous and x is of bounded variation, then∫df(x, S[p](x))dS[p](x) has

finite 1-variation and is equal to (the signature of) the Stieltjes integral∫df(x)dx.

Therefore: (∫df(x, S[p](x))dS[p](x)

)1

u,v

= f(xv)− f(xu)

for all (u, v) ∈ ∆[s,t].(2) S[p](f(x)) has finite 1-variation and S[p](f(x))

1u,v = f(xv)−f(xu) for all (u, v) ∈ ∆[s,t].

Two multiplicative functionals that have finite 1-variation and which terms of the 1st degreeagree are equal by theorem 2.28. Therefore, the sought identity stands.


5. Rough paths and categories

5.1. Elements from category theory. To highlight the minimal framework on which wecan define the notion of rough paths, we will be using the language of categories. We referfor example to [1] for the definitions below which may also be gathered in [23].

Definition 5.1 (Category). A category C is a triple ((Ui)i∈I , (hom(Ui, Uj))i,j∈I , (ψi,j,k)i,j,k∈I)satisfying the three following axioms:

(1) For every i, j ∈ I, hom(Ui, Uj) is a set.(2) For every i, j, k ∈ I, ψi,j,k is a mapping from hom(Ui, Uj)×hom(Uj , Uk) into hom(Ui, Uk).(3) (Associativity) For every i, j, k, l ∈ I, f ∈ hom(Ui, Uj), g ∈ hom(Uj, Uk) and

h ∈ hom(Uk, Ul):

ψi,k,l(ψi,j,k(f, g), h) = ψi,j,l(f, ψj,k,l(g, h))

(4) (Existence of identities) For every i ∈ I, there exists idUi∈ hom(Ui, Ui) such

that, for all j ∈ I, g ∈ hom(Ui, Uj) and h ∈ hom(Uj, Ui), we have:

ψi,i,j(idUi, g) = g and ψj,i,i(h, idUi

) = h

In this case, for all i, j, k ∈ I, Ui is called an object of C; an element in hom(Ui, Uj) (alsodenoted by homC(Ui, Uj)) is called either an arrow, a morphism or a homomorphism in C.The collections (Ui)i∈I and (homC(Ui, Uj))i,j∈I are respectively denoted ob(C) and hom(C).The binary operation ψi,j,k is called a composition of morphisms and is simply denoted by .i.e. for i, j, k ∈ I, f ∈ hom(Ui, Uj) and g ∈ hom(Uj, Uk), ψi,j,k(f, g) is denoted g f .

Remark 5.2. If C = ((Ui)i∈I , (homC(Ui, Uj))i,j∈I , ) is a category, and with the notationsof the previous definition, the associativity and the existence of identities axioms can berewritten in the following way:

h (g f) = (h g) f

andg idUi

= g and idUi h = h

Examples 5.3. • By taking a family of sets considered as objects and all maps betweenthese sets considered as arrows and the composition of maps as binary operations weobtain a category, usually called the category of sets.

• Let (Gi)i∈I be a family of groups. For every i, j ∈ I, let hom(Gi, Gj) be the set of allgroup homomorphisms from Gi to Gj. Then C = ((Gi)i∈I , (hom(Gi, Gj))i,j∈I , ) is acategory (called category of groups).

• Let (Mi)i∈I be a family of topological spaces (respectively smooth manifolds). Forevery i, j ∈ I, let hom(Mi,Mj) be the set of all continuous maps (resp. smoothmaps) from Mi to Mj. Then C = ((Mi)i∈I , (hom(Mi,Mj))i,j∈I , ) is a category.

Definition 5.4. Let C1 and C2 be two categories. A functor (or a functorial rule) from C1to C2 is a pair of mappings ob(C1) → ob(C2) and hom(C1) → hom(C2), which we will denoteby the same letter F , satisfying:

(1) For every object U in C1, F (U) is an object in C2;(2) For every two objects U and V in C1 and an arrow f ∈ homC1(U, V ), F (f) is in

homC2(F (U), F (V ));(3) For all objects U, V and W in C1, and arrows f ∈ homC1(U, V ) and g ∈ homC1(V,W )

F (g f) = F (g) F (f) and F (idU) = idF (U).


Examples 5.5. • Given any category, there exists a trivial functor from this categoryto itself preserving all objects and arrows called the identity functor.

• Given a category of groups C1, let C2 be a category of sets whose objects are theunderlying sets to the objects of C1 and whose arrows are the group homomorphismsin C1 taken as maps between the underlying sets. The natural map associating to eachobject and arrow in C1 their set-counterparts in C2 is a functor. Functors constructedin a similar manner are called forgetful functors.

• Let (Gi)i∈I be a family of Lie groups. For every i, j ∈ I, let hom(Gi, Gj) denote theset of all Lie group homomorphisms from Gi to Gj, Lie(Gi) denote the Lie algebra ofGi and hom(Lie(Gi),Lie(Gj)) denote the set of all Lie algebra homomorphisms fromLie(Gi) to Lie(Gi). Then

C1 := ((Gi)i∈I , (hom(Gi, Gj))i,j∈I , )

and

C2 := ((Lie(Gi))i∈I , (hom(Lie(Gi),Lie(Gj)))i,j∈I , )

are categories. The mapping associating to each object in C1 its Lie algebra and toeach arrow in C1 its pushforward at the identity defines a functor from C1 to C2:

Gi 7→ Lie(Gi); ϕ ∈ hom(Gi, Gj) 7→ ϕ∗(1Gi) ∈ hom(Lie(Gi),Lie(Gj))

5.2. A functorial rule for rough paths. Let E be a Banach space and 1 ≤ p < γ. Let(Ui)i∈I be a family of open subsets of E. It is obvious that

C1 = ((Ui)i∈I , (Liploc(γ, Ui, Uj)))i,j∈I , )

is a category. For i, j ∈ I, we denote by hom(GΩp(Ui), GΩp(Uj)) the set of maps that assignin a continuous way (in the p-rough path topology) to each rough path in Ui a rough pathin Uj defined over the same compact interval. It is also straightforward that:

C2 = ((GΩp(Ui))i∈I , (hom(GΩp(Ui), GΩp(Uj)))i,j∈I , )

defines a category. Finally for any open subsets U and V of E and f ∈ Liploc(γ, U, V ), wedenote by f∗ the following map:

f∗ : GΩp(U) −→ GΩp(V )(x,X, J) 7−→ (f(x),

∫df(x,X)dX, J)

Recall that by theorem 4.23, f∗ : GΩp(J ;U ;E) → GΩp(J ;V ;E) is continuous for any com-pact interval J .

Remark 5.6. In order to use the formalism of category theory, we need the identity mapbetween two open sets to be an arrow. This is one of the reasons we introduce locally Lipschitzmaps.

Theorem 5.7. The rule that assigns to every object U in C1 the object GΩp(U) and to everymorphism f ∈ Liploc(γ, U, V ), where U and V are objects in C1, the map f∗, is functorial.

Proof. For every open subsets U , V and W of E that are objects in C, we need to prove thefollowing:

(1) (IdU)∗ = IdGΩp(U);(2) ∀f ∈ Liploc(γ, U, V ), ∀g ∈ Liploc(γ, V,W ) : (g f)∗ = g∗ f∗.


The first assertion regarding the identity map is straightforward. Let U , V and W be opensubsets of E that are objects in C. Let f ∈ Liploc(γ, U, V ) and g ∈ Liploc(γ, V,W ). Let x bea U -valued path over a segment J with bounded variation. By lemma 4.24

f∗(x, S[p](x), J) = (f(x),

∫df(x, S[p](x))dS[p](x), J) = (f(x), S[p](f(x)), J)

As f(x) has bounded variation (lemma 4.19), we similarly have:

g∗(f(x), S[p](f(x)), J) = (g f(x), S[p](g f(x)), J)

and as g f is locally Lip-γ:

(g f(x), S[p](g f(x)), J) = (g f(x),

∫d(g f)(x, S[p](x))dS[p](x), J)

Therefore ((g f)∗)|GΩ1(U) = (g∗ f∗)|GΩ1(U). As both (g f)∗ and g∗ f∗ are continuous inthe p-variation metric (theorem 4.23) and as GΩ1(U) is dense in GΩp(U) for this metric wededuce that (g f)∗ = g∗ f∗.

5.3. A new definition of rough paths on manifolds. Based on our findings so far, weare now to able to give a minimal approach for defining rough paths on a manifold.

Definition 5.8. Let γ ≥ 1. Let M be a topological manifold and A an atlas on M . We saythat (M,A) is a locally Lip− γ n-manifold if for any two charts (U, φ) and (V, ψ) in A suchthat U ∩ V 6= ∅, the map ψ φ−1 : φ(U ∩ V ) → Rn is locally Lip− γ.

Examples 5.9. • A Lip-γ manifold is a locally Lip-γ manifold. In particular, finite-dimensional vector spaces are locally Lip-γ manifolds.

• Let n ∈ N∗. A Cn manifold is a locally Lip-n manifold.

Definition 5.10. Let n ∈ N∗ and p, γ ∈ [1,+∞[ such that γ > p. Let M be a locallyLip − γ n-manifold. A local p-rough path on M over a compact interval J is a collection(xi, Xi, Ji, (φi, Ui))i∈I of p-rough paths on Rn satisfying the following conditions:

• (Ji)i∈I is a compact cover for J with compact intervals;• For every i ∈ I, (φi, Ui) is a locally Lip− γ chart on M .• ∀i ∈ I : (xi, Xi, Ji) ∈ GΩp(φi(Ui));• (Consistency condition) If i, k ∈ I such that Ji ∩ Jk 6= ∅, then we have:

(φk φ−1i )∗(xi, Xi, Ji ∩ Jk) = (xk, Xk, Ji ∩ Jk)

We identify similar local rough paths in the following way:

Definition 5.11. Let p, γ ∈ [1,+∞[ such that γ > p. Let M be a locally Lip− γ manifoldand J a compact interval. Two local p-rough paths A = (xi, Xi, Ji, (φi, Ui))i∈I and B =(xi, Xi, Ji, (φi, Ui))i∈K on M over J are said to be equivalent if A∪B is also a local p-roughpath.

This, of course, defines an equivalence relation. We will henceforth only consider theequivalence classes associated to this relation.

Definition 5.12. Let p, γ ∈ [1,+∞[ such that γ > p. Let M be a locally Lip− γ manifold.A local p-rough path (xi, Xi, Ji, (φi, Ui))i∈I on a compact interval J is said to be a p-roughpath extension for the path x : J →M if the following holds:

∀i ∈ I : x(Ji) ⊆ Ui and xi = φi x|Ji


If such a rough path exists, we say then that x admits a p-rough path extension.

Following definition 5.11, we will consider that two rough path extensions of a given pathto be the same if their union is a local rough path extension of that path.

5.4. Consistency with previous definitions. We make sure that our notion of roughpath is compatible with the classical one:

Lemma 5.13. Let p, γ ∈ [1,+∞[ such that γ > p. Let M be a Lipschitz-γ manifold and Ja compact interval. Let (xi, Xi, Ji, (φi, Ui))i∈I be a local p-rough path on M over J . Thenthere exists a unique p-rough path X on M such that:

∀i ∈ I : (φi)∗(X)|Ji = (xi, Xi)

Proof. By lemma 4.21 and the definition of a rough path on a manifold, it is clear that if sucha rough path exists, it is necessarily unique. Let (aj)0≤j≤n be a subdivision of J and (ij)0≤j≤n

be a finite sequence of elements of I such that for all j ∈ [[0, n− 1]], we have [aj , aj+1] ⊆ Jij .For j ∈ [[0, n− 1]], we define the rough path:

(yj, Yj) = (xij |[aj ,aj+1], Xij |[aj ,aj+1]

) ∈ GΩp(Uij )

It is trivial then that (yj, Yj, [aj, aj+1], (φij , Uij ))0≤j≤n is equivalent to (xi, Xi, Ji, (φi, Ui))i∈I .

For j ∈ [[0, n − 1]], we define the p-rough path Zj = (φ−1ij)∗(yj(aj), Yj) on M (with starting

point denoted zj). Let X = Z1 ∗ · · ·Zn−1. One can check that the latter concatenation isindeed well-defined as we have, for every j ∈ [[0, n−1]] and f Banach space-valued compactlysupported Lip-γ map on M :

Zj(f∗)1aj ,aj+1

= Yj((f φ−1ij)∗)

1aj ,aj+1

= f φ−1ij(yj(aj+1))− f φ−1

ij(yj(aj)) = f(zj+1)− f(zj)

By construction X satisfies the required conditions.

This identification is in fact one-to-one:

Theorem 5.14. Let p, γ ∈ [1,+∞[ such that γ > p. LetM be a Lipschitz-γ manifold. Thereis a one-to-one mapping between p-rough paths on M and local p-rough paths on M .

Proof. Without loss of generality, we consider rough paths defined over the unit interval J =[0, 1]. Let X be a p-rough path inM over [0, 1] with starting point x. Following theorem 3.17,let (Xi, xi, (φi, Ui))1≤i≤N be a localising sequence for X adapted to a subdivision (si)0≤i≤N of[0, 1]. For i ∈ [[1, N ]], (φi)∗(X

i) identifies with a local geometric p-rough path in φi(Ui) over[si−1, si] that we denote (yi, Yi). We define the collection Y = (yi, Yi, [si−1, si], (φi, Ui))1≤i≤N .For i ∈ [[1, N − 1]], we have, first by the identification between (yi, Yi) and (Xi, xi), then theconsistency of the endpoint of Xi with the starting point of Xi+1:

(φi+1 φ−1i )∗(yi, Yi, si) = (φi+1 φ

−1i )∗(yi(si), 1, si)

= ((φi+1 φ−1i )(yi(si)), 1, si)

= ((φi+1 φ−1i )(φi(xi) + (Xi(φ−1

i )∗)1si−1,si

, 1, si)= (φi(xi+1), 1, si)= (yi+1, Yi+1, si)

This proves that Y is indeed a local p-rough path on M over [0, 1]. Lemma 5.13 shows thatthe mapping X 7→ Y constructed above is onto and one-to-one.


6. Coloured paths on manifolds

There is a general principle underlying the procedure followed above that can be usedto define any notion of colouring already existing on the Euclidean space to a manifold,assuming that we can find a suitable functorial rule. The lift of a path into a rough path canbe indeed seen as a colouring: an extra bit of information that cannot necessarily be learnedby looking at the base path only.

We can define a notion of coloured charts and atlas over any n-topological manifold pro-vided that the transition maps are arrows in an appropriate category. We will call such amanifold a coloured manifold.

Definition 6.1. Let n ∈ N∗. Let C be a category whose objects are all open subsets ofRn and for which inclusion and restriction maps are also arrows. Let M be a topologicalmanifold and A an atlas on M . We say that (M,A) is a coloured manifold with respect toC if for any two charts (U, φ) and (V, ψ) in A such that U ∩ V 6= ∅, we have ψ φ−1 ∈homC(φ(U ∩ V ), ψ(U ∩ V )).

We easily retrieve some familiar constructions of manifolds using this language:

Example 6.2. Let n ∈ N∗. Let C1 be the category whose objects are all open subsets ofRn and whose arrows are continuous maps between these sets. Topological n-dimensionalmanifolds are exactly the coloured manifolds with respect to C1.

Example 6.3. Let n ∈ N∗. Let S1 be the category whose objects are all open subsets of Rn

and whose arrows are smooth maps between these sets. Smooth n-dimensional manifolds arethe coloured manifolds with respect to S1.

Example 6.4. Let n ∈ N∗ and γ ≥ 1. Let L1 be the category whose objects are all opensubsets of Rn and whose arrows are locally Lipschitz-γ maps between these sets. LocallyLip-γ n-dimensional manifolds are the coloured manifolds with respect to L1.

Suppose now that we have a notion of coloured paths on Rn that have underlying basepaths which we will call traces (rough paths are an example). Denote by T (U) the sets ofcoloured paths whose traces lie in an open subset U of Rn. Denote by C a category as in

definition 6.1 and let C be a category whose objects are the sets of coloured paths with traces

in open subsets of Rn. We assume there exists a functor from C to C:

U 7→ T (U) ; f 7→ f∗

such that for each arrow f between objects U and V of C, the arrow f∗ between T (U) andT (V ) associates to each coloured path X on U a coloured path on V whose trace is theimage by f of the trace of X . We can now define coloured paths on a coloured manifoldin the same manner as in definition 5.10 and the existence of coloured path extensions formanifold-based paths as in definition 5.12.

Definition 6.5. Let n ∈ N∗. Let C be a category whose objects are all open subsets of Rn andfor which inclusion and restriction maps are also arrows. Let M be a coloured n-manifoldw.r.t C. A coloured path on M over a compact interval J is a collection (Xi, Ji, (φi, Ui))i∈Iof coloured paths on Rn satisfying the following conditions:

• (Ji)i∈I is a compact cover for J by segments;• For every i ∈ I, (φi, Ui) is in the atlas of the coloured manifold M .• ∀i ∈ I : Xi ∈ T (φi(Ui)) and Xi is defined over Ji;


• (Consistency condition) If i, k ∈ I such that Ji ∩ Jk 6= ∅, then we have:

(φk φ−1i )∗(Xi|Ji∩Jk) = Xk |Ji∩Jk

A coloured path (Xi, Ji, (φi, Ui))i∈I on an interval J is said to be a coloured path extensionfor the path x : J →M if the following holds:

∀i ∈ I : x(Ji) ⊆ Ui and trace(Xi) = φi x|Ji

If such a coloured path exists, we say then that x admits a coloured path extension.

As the study of the classical examples below will show, we emphasize that the rule linking C

and C need be functorial in order to have sound geometric definitions and for these definitionsto make sense on their own and be consistent with the definitions of coloured paths on theEuclidean space seen now as a coloured manifold.

Example 6.6 (Example 6.2 continued). Take C2 to be the category whose objects are thesets of continuous paths with values in open subsets of Rn. The colouring map associatedto an arrow f in C1 is a map that assigns to every continuous path x the path f x. Thenour new definition of a continuous path on M (as a coloured path) can be identified with theclassical one which relies only on the topology on M : every continuous path on M in theclassical sense can be seen as the concatenation of pushforwards of continuous paths on theEuclidean space that are consistent among themselves.

Example 6.7 (Example 6.3 continued). Take S2 to be the category whose objects are thesets of smooth paths valued in open subsets of Rn. The associated functorial rule from S1 toS2 is the same as above by replacing continuity with smoothness. Finally, one can see thedefinitions of smooth maps in the classical sense and when described as coloured paths areequivalent.

Example 6.8 (Example 6.4 continued). Let 1 ≤ p < γ. Let L2 be the category whoseobjects are sets of local geometric p-rough paths supported in the open subsets of Rn andwhose arrows are the set of mappings between the rough path in these objects defined over thesame interval in a continuous way (in the p-rough path topology). We showed in the previoussections that rough paths can be seen as coloured paths in this setting.

References

[1] Adamek, J., Herrlich, H., and Strecker, G.E. (1990) Abstract and Concrete Categories – The Joy of

Cats. John Wiley & Sons Inc.[2] Arribas, P.I., Goodwin, G.M., Geddes, J.R., Lyons, T.J. and Saunders, K. (2018) A signature-based

machine learning model for distinguishing bipolar disorder and borderline personality disorder. TranslPsychiatry, issue 1 volume 8 page 274.

[3] Bailleul, I. (2014) Rough integrators on Banach manifolds arXiv:1403.3285.[4] Boedihardjo, H., Geng, X., Lyons, T.J. and Yang, D. (2016) The signature of a rough path: Uniqueness.

Advances in Mathematics,720-737.[5] Boutaib, Y. (2015) On Lipschitz maps and the Holder regularity flows. arXiv:1510.07614 (due to be

published in Rev. Roumaine Math. Pures Appli.).[6] Caruana, M., Levy T. and Lyons, T.J. (2007) Differential Equations Driven by Rough Paths. Volume

1908 of Lecture Notes in Mathematics. Springer, Berlin.[7] Cass, T., Driver, B.K. and Litterer, C. (2015) Constrained Rough Paths. Proceedings of the London

Mathematical Society, 111: 1471-1518.[8] Cass, T., Litterer, C and Lyons, T.J. (2012) Rough paths on manifolds. Interdiscip. Math. Sci. 12,

33–88.

http://arxiv.org/abs/1403.3285



[9] Chen, K. (1957) Integration of paths , geometric invariants and a generalized Baker-Hausdorff formula.Ann. of Math. (2) 65, 163-178.

[10] Chen, K. (1958) Integration of paths - a faithful representation of paths by non-commutative formalpower series. Trans. Am. Math. Soc. 89, 395–407.

[11] Chevyrev, I., Nanda, V. and Oberhauser, H. (2018) Persistence paths and signature features in topo-logical data analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 11.

[12] Chistyakov, V.V., Galkin, O.E. (1998): On maps of bounded p-variations with p > 1. Positivity, 2,1945. 15

[13] Driver, B.K. and Semko, J.S. Controlled Rough Paths on Manifolds I (2017), Rev. Mat. Iberoam. 33,no. 3, 85950.

[14] Friz, P.K. and Hairer, M. (2014) A Course on Rough Paths. With an introduction to Regularity Struc-

tures. Universitext, p. xiv+251. Springer, New York.[15] Friz, P.K. and Victoir N.B. (2010) Multidimensional stochastic processes as rough paths, vol. 120 of

Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. Theoryand applications.

[16] Gubinelli, M. (2004) Controlling rough paths. J. Funct. Anal. 216, no. 1, 86–140.[17] Gubinelli, M. (2010) Ramification of rough paths. J. Differential Equations 248, no. 4, 693-721.[18] Gubinelli, M., Imkeller, P. and Perkowski, N. (2012) Paraproducts, rough paths and controlled distribu-

tions. arXiv:1210.2684.[19] Graham, B. (2013) Sparse arrays of signatures for online character recognition. arXiv:1308.0371.[20] Hairer, M. (2014) A theory of regularity structures. Invent. Math 198, no. 2, 269–504.[21] Hambly, B. and Lyons, T.J. (2010) Uniqueness for the signature of a path of bounded variation and the

reduced path group. Ann. of Math. 171, no. 1, 109-167.[22] Le Jan, Y. and Qian, Z. Stratonovich’s signatures of Brownian motion determine Brownian sample

paths. Probab. Theory Related Fields 157 (2013), no. 1-2, 209223.[23] Lee, J.M. (2002) Introduction to Smooth Manifolds. Graduate Texts in Mathematics, Springer.[24] Lyons, T. (1994). Differential equations driven by rough signals. I. An extension of an inequality by

L.C.Young. Mathematical Research Letters 1, 451464.[25] Lyons, T.J. (1998) Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14, no. 2,

215–310.[26] Lyons, T.J. and Qian, Z. (2002) System control and rough paths. Oxford Mathematical Monographs.

Oxford University Press, Oxford Science Publications, Oxford.[27] Lyons, T.J and Nicolas Victoir, N. (2007) An extension theorem to rough paths. Ann. Inst. H. Poincar

Anal. Non Linaire 24, no. 5, 835–847.[28] Reutenauer, C. Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford

Science Publications. The Clarendon Press, Oxford University Press, New York, 1993.[29] Stein, E.M. (1970) Singular integrals and differentiability properties of functions. Princeton University

Press.[30] Young, L.C. (1936) An inequality of Holder type connected with Stieltjes integration Acta Math. 67,

251–282.

RWTH Aachen University. Chair for Mathematics of Information Processing. Pontdri-

esch 10, 52062 Aachen. Germany.

E-mail address : [email protected]

Mathematical Institute. University of Oxford. Andrew Wiles Building. Radcliffe Ob-

servatory Quarter, Woodstock Road. Oxford OX2 6GG. United Kingdom.

E-mail address : [email protected]



arxiv:1510.07833v2 [math.ca] 28 oct 2015arxiv:1510.07833v2 [math.ca] 28 oct 2015 a new definition of...

Documents